Difference between revisions of "T-Channel"

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<center><math>\textbf{\underline{Navigation}}</math>
+
<center><math>\underline{\textbf{Navigation}}</math>
  
 
[[S-Channel|<math>\vartriangleleft </math>]]
 
[[S-Channel|<math>\vartriangleleft </math>]]
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The t quantity is known as the square of the 4-momentum transfer
 
The t quantity is known as the square of the 4-momentum transfer
  
<center><math>t \equiv \left({\mathbf P_1^*}- {\mathbf P_1^{'*}}\right)^2=\left({\mathbf P_2^{*}}+ {\mathbf P_2^{'*}}\right)^2</math></center>
+
<center><math>t \equiv \left({\mathbf P_1^*}- {\mathbf P_1^{'*}}\right)^2=\left({\mathbf P_2^{*}}- {\mathbf P_2^{'*}}\right)^2</math></center>
  
 
<center>[[File:400px-CMcopy.png]]</center>
 
<center>[[File:400px-CMcopy.png]]</center>
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In the center of mass frame of reference,  
 
In the center of mass frame of reference,  
  
<center><math> E_1^*=E_1^{'*} \quad E_2^*=E_2^{'*} \quad E_1^*=E_2^*</math></center>
+
<center><math> E^* \equiv E_1^*=E_1^{'*} = E_2^*=E_2^{'*} = E_1^*=E_2^*</math></center>
  
  
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<center><math>| \vec p \ _1^*|=| \vec p \ _1^{'*}| \quad | \vec p \ _2^*|=| \vec p \ _2^{'*}|</math></center>
+
<center><math>|p^*| \equiv | \vec p \ _1^*|=| \vec p \ _1^{'*}| =| \vec p \ _2^*|=| \vec p \ _2^{'*}|</math></center>
  
  
 +
and <math>\theta_1</math> is the angle between <math>\vec p \ _1^* </math> and <math> \vec p \ _1^{'*}</math>
  
<center><math>t \equiv 2m_1^*-2E_1^{*2}+2 p \ _1^{*2}cos\ \theta</math></center>
+
 
 +
 
 +
<center><math>t \equiv 2m_1^*-2E_1^{*2}+2   |p |^{*2}cos\ \theta</math></center>
  
  
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<center><math>t \equiv -2 p \ _1^{*2}(1-cos\ \theta)</math></center>
+
<center><math>t \equiv -2 p \ ^{*2}(1-cos\ \theta)</math></center>
  
  
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<center><math>\textbf{\underline{Navigation}}</math>
+
<center><math>\underline{\textbf{Navigation}}</math>
  
 
[[S-Channel|<math>\vartriangleleft </math>]]
 
[[S-Channel|<math>\vartriangleleft </math>]]

Latest revision as of 18:49, 15 May 2018

[math]\underline{\textbf{Navigation}}[/math]

[math]\vartriangleleft [/math] [math]\triangle [/math] [math]\vartriangleright [/math]


t Channel

The t quantity is known as the square of the 4-momentum transfer

[math]t \equiv \left({\mathbf P_1^*}- {\mathbf P_1^{'*}}\right)^2=\left({\mathbf P_2^{*}}- {\mathbf P_2^{'*}}\right)^2[/math]
400px-CMcopy.png


[math]t \equiv \left({\mathbf P_1^*}- {\mathbf P_1^{'*}}\right)^2[/math]


[math]t \equiv \mathbf P_1^{*2}-2 \mathbf P_1^* \mathbf P_1^{'*}+ \mathbf P_1^{'*2}[/math]


[math]t \equiv 2m_1^2-2E_1^*E_1^{'*}+2 \vec p \ _1^* \vec p \ _1^{'*}[/math]


In the center of mass frame of reference,

[math] E^* \equiv E_1^*=E_1^{'*} = E_2^*=E_2^{'*} = E_1^*=E_2^*[/math]


and


[math]|p^*| \equiv | \vec p \ _1^*|=| \vec p \ _1^{'*}| =| \vec p \ _2^*|=| \vec p \ _2^{'*}|[/math]


and [math]\theta_1[/math] is the angle between [math]\vec p \ _1^* [/math] and [math] \vec p \ _1^{'*}[/math]


[math]t \equiv 2m_1^*-2E_1^{*2}+2 |p |^{*2}cos\ \theta[/math]


Using the relativistic term for Energy


[math]E^2=\vec p \ ^2+m^2[/math]


[math]t \equiv -2 p \ ^{*2}(1-cos\ \theta)[/math]




[math]\underline{\textbf{Navigation}}[/math]

[math]\vartriangleleft [/math] [math]\triangle [/math] [math]\vartriangleright [/math]